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We prove a triangular decomposition theorem for the lower crystal lattice $\\OAztG$ of the quantized function algebra $\\OtG$, establishing that $\\OAztG=A_0\\text{-alg}<\\RAzp \\cup \\RAzm>.$ This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\\cite{DDPa} to all simple complex Lie algebras. As a consequence, we obtain: (i) the inclusion $\\OAztG\\subs"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2603.21868","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2026-03-23T12:02:02Z","cross_cats_sorted":["math.OA","math.RT"],"title_canon_sha256":"a6fe59c95b0022370039b39df325dd40d39c965f7320b6b7d76d11066a2e32ed","abstract_canon_sha256":"809969ea0c9133fb4a35cea62dea46848a921594b6dd896eb8244bda57da47f0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:12:53.388139Z","signature_b64":"PAkSeLhe0AF3IWC5ddGuqir0eyMulrv50Ag/+V+saUsqV+/MPj6dvxkWEL0lB2g6fTL/MmiTemDf1Ds07u98Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a457330d4a379079b4ca009f3e639e9c8e5a604739a6a4a711b5a0ca3ae8cedb","last_reissued_at":"2026-06-19T16:12:53.387688Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:12:53.387688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Triangular Decomposition of the Crystal Lattice of Quantized Function Algebras: Revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA","math.RT"],"primary_cat":"math.QA","authors_text":"Ayan Dey","submitted_at":"2026-03-23T12:02:02Z","abstract_excerpt":"Let $\\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected complex Lie group with $\\mathrm{Lie}(G)=\\g$ and compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\\OAztG$ of the quantized function algebra $\\OtG$, establishing that $\\OAztG=A_0\\text{-alg}<\\RAzp \\cup \\RAzm>.$ This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\\cite{DDPa} to all simple complex Lie algebras. 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